PAT data analysis

Task

4 different FBs = -5, -1, +1, +5
Participants can either click after seeing the cue: ‘Hit’, or not do anything ‘Miss’. In both cases they then see the reward but only recieve it if Hit. Frequencies of reward are cue-specific.
Cues = High Punishment (Cue_HP), Low Punishment (Cue_LP), Low Reward (Cue_LR), High Reward (Cue_HR)
N trials = 112 (per cue = 28)
N runs = 2 (56 trials per run, participants had a short break in between)

Reward frequencies:

##       Cue  R_-5  R_-1   R_1   R_5
## 0  Cue_HP  14.0   6.0   4.0   4.0
## 1  Cue_HR   4.0   4.0   6.0  14.0
## 2  Cue_LP  10.0  10.0   4.0   4.0
## 3  Cue_LR   4.0   4.0  10.0  10.0

Task information

Behaviour

Overall   N participants = 170.

Summary of Behaviour

Accross time
Participant average:
Bump at the start of the 2nd run (each run is made of 56 trials)
t3 = \(3*16\) = 48 trials
t4 = \(4*16\) = 64 trials

Behaviour accross time

Per Run
Doesn’t seem to be an overall difference between runs

Behaviour by run

Per Block
Pressing bias at the beginning of each run

Behaviour by run

Simulations

Plot softmax for different betas in a likely value range [-10,10] to know how to constrict the beta parameter in the model fitting.

Softmax visualisation

Functions

Model glossary: \(PE\) = prediction error, \(FB\) = observed feedback (irrespective of hit), \(V^{miss}\) = 0

Value functions

Rescorla Wagner no V0

function name = rescorla_wagner_noV0
rescorla_wagner with fixed parameter: \(V_0\) = 0

Rescorla Wagner reinitialising V0

function name = rescorla_wagner
For each cue:
If t = 56:
\[ V_{t} = V_0 \] else:
\[ PE = FB_{t} - V_{t}^{hit} \] \[ V_{t+1}^{hit} = V_{t}^{hit} + \alpha \cdot PE \]

Rescorla Wagner

function name = rescorla_wagner
For each cue:
\[ PE = FB_{t} - V_{t}^{hit} \] \[ V_{t+1}^{hit} = V_{t}^{hit} + \alpha \cdot PE \]

Rescorla Wagner 2 learning rates

function name = rescorla_wagner_2LR_FB
For each cue: \[ PE = FB_{t} - V_{t}^{hit} \] FB could take the following values: -5, -1, +1, +5
Different learning rates for reward and punishment:
if \(FB_t\) > 0: \[ V_{t+1}^{hit} = V_{t}^{hit} + \alpha_{rew} \cdot PE \] if \(FB_t\) < 0: \[ V_{t+1}^{hit} = V_{t}^{hit} + \alpha_{pun} \cdot PE \]

Rescorla Wagner weighted FB

function name = rescorla_wagner_weightRew
For each cue:
Scaling of feedback:
if abs(\(FB_t\)) = 5:
\[ FB_{t} = w \cdot FB_{t} \] Prediction error:
\[ PE = FB_{t} - V_{t}^{hit} \] \[ V_{t+1}^{hit} = V_{t}^{hit} + \alpha \cdot PE \]

Rescorla Wagner shrinking learning rate

function name = rescorla_wagner_shrinking_alpha
For each cue:
\[ PE = FB_{t} - V_{t}^{hit} \] Shrinking factor: \[ shrink = \frac{N_{trials} - t}{N_{trials}} \] With \(N_{trials}=112\), and \(t \in [1, 112]\)
\[ V_{t+1}^{hit} = V_{t}^{hit} + \alpha_t \cdot shrink \cdot PE \]

rescorla_wagner_reinitV0

Decision functions

Softmax

function name = my_softmax
For each cue: \[ p_{t}(hit) = \frac {e^{\beta \cdot V_{t}^{hit}}}{e^{\beta \cdot V_{t}^{hit}}+e^{\beta \cdot V^{miss}}} = \frac{e^{\beta \cdot V_{t}^{hit}}}{e^{\beta \cdot V_{t}^{hit}}+1} \]

Softmax press bias

function name = my_softmax_press_bias
For each cue: \[ p_{t}(hit) = \frac{e^{\beta \cdot (V_{t}^{hit}+\pi)}}{e^{\beta \cdot (V_{t}^{hit}+\pi)}+e^{\beta \cdot V^{miss}}} = \frac{e^{\beta \cdot (V_{t}^{hit}+\pi)}}{e^{\beta \cdot (V_{t}^{hit}+\pi)}+1} \]

Softmax shrinking press bias

function name = my_softmax_shrinking_press_bias
Shrinking factor: \[ shrink = \frac{N_{run trials} - t_{run}}{N_{run trials}} \] With \(N_{runtrials}=56\), and \(t_{run} \in [1, 56]\)

For each cue: \[ p_{t}(hit) = \frac{e^{\beta \cdot (V_{t}^{hit}+\pi_t \cdot shrink)}}{e^{\beta \cdot (V_{t}^{hit}+\pi_t \cdot shrink)}+e^{\beta \cdot V^{miss}}} = \frac{e^{\beta \cdot (V_{t}^{hit}+\pi_t \cdot shrink)}}{e^{\beta \cdot (V_{t}^{hit}+\pi_t \cdot shrink)}+1} \]

Models

Model 0: alpha, beta

mod = 0
mod_info = print_model_info(mod)

## Model = model0
## Value function = rescorla_wagner_noV0
## Decision function = my_softmax
## Parameters = alpha, beta

Free parameters: \(\alpha\) = learning rate, \(\beta\) = inverse temperature
Fixed parameter: \(V_0 = 0\)

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams         alpha          beta
## count  170.000000    170.0    170.0  1.700000e+02  1.700000e+02
## mean    58.107500    112.0      2.0  6.582904e-02  6.519460e+00
## std     15.140554      0.0      0.0  9.337897e-02  6.652688e+00
## min      5.050840    112.0      2.0  3.737686e-09  5.820973e-09
## 25%     48.714325    112.0      2.0  4.740678e-03  1.290706e+00
## 50%     61.159156    112.0      2.0  1.462594e-02  3.345980e+00
## 75%     70.869630    112.0      2.0  1.136493e-01  9.997270e+00
## max     77.632484    112.0      2.0  5.801689e-01  2.000000e+01

Plots

Model parameters

Model predictions

Model 1: alpha, beta, v0

mod = 1
mod_info = print_model_info(mod)

## Model = model1
## Value function = rescorla_wagner
## Decision function = my_softmax
## Parameters = v0, alpha, beta

Free parameters: \(\alpha\) = learning rate, \(\beta\) = inverse temperature, \(V_0\) = prior mean

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams          v0         alpha        beta
## count  170.000000    170.0    170.0  170.000000  1.700000e+02  170.000000
## mean    52.360935    112.0      3.0    0.939961  9.182191e-02    5.850742
## std     15.228512      0.0      0.0    2.489884  1.355430e-01    6.377281
## min      4.867525    112.0      3.0  -10.000000  4.399281e-17    0.089751
## 25%     42.300949    112.0      3.0    0.025935  8.848244e-03    0.980210
## 50%     54.590413    112.0      3.0    0.198867  3.341520e-02    2.523140
## 75%     64.096722    112.0      3.0    1.003701  1.231332e-01    8.274388
## max     76.499244    112.0      3.0   10.000000  7.821228e-01   20.000000

Plots

Model parameters

Model predictions

Model 2: alpha, beta, v0, pi

mod = 2
mod_info = print_model_info(mod)

## Model = model2
## Value function = rescorla_wagner
## Decision function = my_softmax_press_bias
## Parameters = v0, alpha, beta, pi

Free parameters: \(\alpha\) = learning rate, \(\beta\) = inverse temperature, \(V_0\) = prior mean, \(\pi\) = press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    50.262414    112.0      4.0
## std     15.314180      0.0      0.0
## min      3.240895    112.0      4.0
## 25%     39.246651    112.0      4.0
## 50%     52.002387    112.0      4.0
## 75%     61.634458    112.0      4.0
## max     76.216917    112.0      4.0
##                v0         alpha        beta          pi
## count  170.000000  1.700000e+02  170.000000  170.000000
## mean     0.990495  1.153701e-01    4.880064    0.051348
## std      2.795047  1.520297e-01    5.862168    1.212308
## min    -10.000000  2.849998e-16    0.100666   -3.717110
## 25%     -0.298361  1.232139e-02    0.910771   -0.641916
## 50%      0.340262  6.827151e-02    2.096763    0.123916
## 75%      1.436749  1.472323e-01    6.459079    0.747604
## max     10.000000  1.000000e+00   20.000000    4.374439

Plots

Model parameters

Model predictions

Model 3: alpha_rew, alpha_pun, beta, v0

mod = 3
mod_info = print_model_info(mod)

## Model = model3
## Value function = rescorla_wagner_2LR_FB
## Decision function = my_softmax
## Parameters = v0, alpha_rew, alpha_pun, beta

Free parameters: \(\alpha_{rew, pun}\) = learning rate for reward/punishment, \(\beta\) = inverse temperature, \(V_0\) = prior mean

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    50.796260    112.0      4.0
## std     15.403051      0.0      0.0
## min      2.906027    112.0      4.0
## 25%     39.812044    112.0      4.0
## 50%     52.587297    112.0      4.0
## 75%     63.267533    112.0      4.0
## max     76.348831    112.0      4.0
##                v0     alpha_rew     alpha_pun        beta
## count  170.000000  1.700000e+02  1.700000e+02  170.000000
## mean     1.191445  1.392805e-01  1.336050e-01    2.581129
## std      2.746771  1.560690e-01  1.659156e-01    3.604441
## min    -10.000000  8.300815e-21  5.344292e-17    0.093321
## 25%      0.072541  4.538379e-02  3.665528e-02    0.648320
## 50%      0.383772  8.588038e-02  7.920729e-02    1.442147
## 75%      1.284494  1.851164e-01  1.452102e-01    2.820631
## max     10.000000  1.000000e+00  1.000000e+00   20.000000

Stats on parameters

# Paired samples t-test
stats.ttest_rel(data_mod_num['alpha_rew'], data_mod_num['alpha_pun'])

## Ttest_relResult(statistic=0.4463957230934856, pvalue=0.6558828708733997)

Plots

Model parameters

Model predictions

Model 4: alpha, beta, v0, w

mod = 4
mod_info = print_model_info(mod)

## Model = model4
## Value function = rescorla_wagner_weightRew
## Decision function = my_softmax
## Parameters = v0, alpha, beta, w

Free parameters: \(\alpha\) = learning rate, \(\beta\) = inverse temperature, \(V_0\) = prior mean, \(\pi\) = press bias
Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    52.822755    112.0      4.0
## std     14.883087      0.0      0.0
## min      5.879380    112.0      4.0
## 25%     42.537941    112.0      4.0
## 50%     54.503453    112.0      4.0
## 75%     64.818023    112.0      4.0
## max     76.690227    112.0      4.0
##                v0         alpha        beta           w
## count  170.000000  1.700000e+02  170.000000  170.000000
## mean     0.978360  9.240670e-02    5.932748    1.999998
## std      2.627040  1.413147e-01    6.605705    0.000029
## min    -10.000000  3.254914e-16    0.060053    1.999627
## 25%      0.025435  1.157620e-02    0.884595    2.000000
## 50%      0.190074  3.639225e-02    2.804845    2.000000
## 75%      0.896349  1.154552e-01    8.564268    2.000000
## max     10.000000  9.698856e-01   20.000000    2.000000

Plots

Model parameters

Model predictions

Model 5: alpha_t, beta, v0

mod = 5
mod_info = print_model_info(mod)

## Model = model5
## Value function = rescorla_wagner_shrinking_alpha
## Decision function = my_softmax
## Parameters = v0, alpha_t, beta

Free parameters: \(\alpha_t\) = shrinking learning rate, \(\beta\) = inverse temperature, \(V_0\) = prior mean

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams          v0       alpha_t        beta
## count  170.000000    170.0    170.0  170.000000  1.700000e+02  170.000000
## mean    51.755054    112.0      3.0    1.041759  1.366831e-01    5.308782
## std     15.374300      0.0      0.0    2.711148  1.721451e-01    5.995968
## min      4.647453    112.0      3.0  -10.000000  2.583236e-16    0.093505
## 25%     41.106491    112.0      3.0    0.025068  1.185567e-02    0.951547
## 50%     54.004206    112.0      3.0    0.244133  6.695768e-02    1.921968
## 75%     63.455126    112.0      3.0    1.305117  1.951048e-01    8.745747
## max     76.580572    112.0      3.0   10.000000  9.189379e-01   20.000000

Plots

Model parameters

Model predictions

Model 6: alpha_rew, alpha_pun, beta, v0, pi_t

mod = 6
mod_info = print_model_info(mod)

## Model = model6
## Value function = rescorla_wagner_2LR_FB
## Decision function = my_softmax_shrinking_press_bias
## Parameters = v0, alpha_rew, alpha_pun, beta, pi_t

Free parameters: \(\alpha_{rew, pun}\) = learning rate for reward/punishment, \(\beta\) = inverse temperature, \(V_0\) = prior mean, \(\pi_t\) = shrinking press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    46.913267    112.0      5.0
## std     16.729136      0.0      0.0
## min      3.155677    112.0      5.0
## 25%     35.609496    112.0      5.0
## 50%     48.677384    112.0      5.0
## 75%     60.739674    112.0      5.0
## max     76.499753    112.0      5.0
##                v0     alpha_rew     alpha_pun        beta        pi_t
## count  170.000000  1.700000e+02  1.700000e+02  170.000000  170.000000
## mean    -0.107151  1.098816e-01  1.311821e-01    3.791271    1.112207
## std      1.964040  1.420514e-01  1.940812e-01    4.530921    1.677841
## min     -5.000000  1.744307e-22  3.737089e-09    0.120519   -7.095337
## 25%     -1.046358  2.654594e-02  2.467444e-02    0.958264    0.233941
## 50%     -0.318710  5.443208e-02  5.757258e-02    1.912919    0.833557
## 75%      0.405973  1.398089e-01  1.516415e-01    4.395514    1.759054
## max      5.000000  1.000000e+00  1.000000e+00   15.000000    7.500008

Stats on parameters

# Paired samples t-test
stats.ttest_rel(data_mod_num['alpha_rew'], data_mod_num['alpha_pun'])

## Ttest_relResult(statistic=-1.7422330810432594, pvalue=0.08328687759338284)

Plots

Model parameters

Model predictions

Model 7: alpha, beta, v0, w, pi_t

mod = 7
mod_info = print_model_info(mod)

## Model = model7
## Value function = rescorla_wagner_weightRew
## Decision function = my_softmax_shrinking_press_bias
## Parameters = v0, alpha, beta, w, pi_t

Free parameters: \(\alpha\) = learning rate, \(\beta\) = inverse temperature, \(V_0\) = prior mean, \(w\) = large FB weight, \(\pi_t\) = shrinking press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    49.086785    112.0      5.0
## std     16.379621      0.0      0.0
## min      5.694755    112.0      5.0
## 25%     37.089098    112.0      5.0
## 50%     50.825920    112.0      5.0
## 75%     62.037113    112.0      5.0
## max     76.516800    112.0      5.0
##                v0         alpha        beta             w        pi_t
## count  170.000000  1.700000e+02  170.000000  1.700000e+02  170.000000
## mean    -0.076820  1.185178e-01    3.839539  2.000000e+00    1.108973
## std      2.130397  1.423396e-01    4.262047  4.173539e-10    2.071525
## min     -5.000000  4.686312e-16    0.123766  2.000000e+00   -5.600134
## 25%     -0.870512  2.929666e-02    0.788073  2.000000e+00    0.127697
## 50%     -0.262699  7.865536e-02    2.308973  2.000000e+00    0.657731
## 75%      0.379249  1.546136e-01    4.779899  2.000000e+00    1.782952
## max      5.000000  1.000000e+00   15.000000  2.000000e+00    8.659615

Plots

## <string>:11: RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (`matplotlib.pyplot.figure`) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam `figure.max_open_warning`).

Model parameters

Model predictions

Model 8: alpha_t, beta, v0, pi_t

mod = 8
mod_info = print_model_info(mod)

## Model = model8
## Value function = rescorla_wagner_shrinking_alpha
## Decision function = my_softmax_shrinking_press_bias
## Parameters = v0, alpha_t, beta, pi_t

Free parameters: \(\alpha_t\) = shrinking learning rate, \(\beta\) = inverse temperature, \(V_0\) = prior mean, \(\pi_t\) = shrinking press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    47.798468    112.0      4.0
## std     16.920126      0.0      0.0
## min      3.325319    112.0      4.0
## 25%     37.156746    112.0      4.0
## 50%     50.614676    112.0      4.0
## 75%     61.386058    112.0      4.0
## max     76.705921    112.0      4.0
##                v0       alpha_t        beta        pi_t
## count  170.000000  1.700000e+02  170.000000  170.000000
## mean    -0.116659  1.811305e-01    3.293070    1.062610
## std      2.272556  2.089355e-01    3.500643    1.764987
## min     -5.000000  1.264101e-16    0.122773   -6.415641
## 25%     -1.234541  3.692270e-02    0.877272    0.192999
## 50%     -0.338046  1.023679e-01    1.757615    0.948407
## 75%      0.635814  2.453132e-01    4.711434    1.927004
## max      5.000000  1.000000e+00   15.000000    6.668164

Plots

Model parameters

Model predictions

Model 9: alpha, beta, v0, pi_t

mod = 9
mod_info = print_model_info(mod)

## Model = model9
## Value function = rescorla_wagner
## Decision function = my_softmax_shrinking_press_bias
## Parameters = v0, alpha, beta, pi_t

Free parameters: \(\alpha\) = learning rate, \(\beta\) = inverse temperature, \(V_0\) = prior mean, \(\pi_t\) = shrinking press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    48.435748    112.0      4.0
## std     16.706529      0.0      0.0
## min      3.563870    112.0      4.0
## 25%     36.917690    112.0      4.0
## 50%     50.649393    112.0      4.0
## 75%     61.727837    112.0      4.0
## max     77.041330    112.0      4.0
##                v0         alpha        beta        pi_t
## count  170.000000  1.700000e+02  170.000000  170.000000
## mean    -0.092623  1.139156e-01    3.445958    1.030806
## std      2.170045  1.427369e-01    3.950705    2.004097
## min     -5.000000  3.737686e-09    0.038199   -7.720410
## 25%     -1.136027  2.639210e-02    0.818287    0.100299
## 50%     -0.309904  6.618056e-02    1.717946    0.984759
## 75%      0.472413  1.488237e-01    4.586511    1.948065
## max      5.000000  1.000000e+00   15.000000    8.429271

Plots

Model parameters

Model predictions

Model 10: alpha_rew, alpha_pun, beta, pi_t

mod = 10
mod_info = print_model_info(mod)

## Model = model10
## Value function = rescorla_wagner_2LR_FB_noV0
## Decision function = my_softmax_shrinking_press_bias
## Parameters = alpha_rew, alpha_pun, beta, pi_t

Free parameters: \(\alpha_{rew, pun}\) = learning rate for reward/punishment, \(\beta\) = inverse temperature, \(\pi_t\) = shrinking press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    49.505559    112.0      4.0
## std     15.829129      0.0      0.0
## min      3.120417    112.0      4.0
## 25%     37.229115    112.0      4.0
## 50%     51.263381    112.0      4.0
## 75%     62.699845    112.0      4.0
## max     76.995867    112.0      4.0
##           alpha_rew     alpha_pun        beta        pi_t
## count  1.700000e+02  1.700000e+02  170.000000  170.000000
## mean   1.266182e-01  1.532147e-01    2.853361    0.990701
## std    1.555312e-01  1.921104e-01    3.664782    1.379024
## min    3.094925e-16  3.465784e-09    0.128481   -3.675747
## 25%    3.569697e-02  3.972559e-02    0.820286    0.234678
## 50%    7.044992e-02  9.065113e-02    1.454570    0.675882
## 75%    1.605202e-01  1.925063e-01    3.103434    1.409141
## max    1.000000e+00  1.000000e+00   15.000000    5.000000

Plots

Model parameters

Model predictions

Model 11: alpha_t, beta, pi_t

mod = 11
mod_info = print_model_info(mod)

## Model = model11
## Value function = rescorla_wagner_shrinking_alpha_noV0
## Decision function = my_softmax_shrinking_press_bias
## Parameters = alpha_t, beta, pi_t

Free parameters: \(\alpha_t\) = shrinking learning rate, \(\beta\) = inverse temperature, \(\pi_t\) = shrinking press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams       alpha_t        beta        pi_t
## count  170.000000    170.0    170.0  1.700000e+02  170.000000  170.000000
## mean    51.128443    112.0      3.0  2.156569e-01    2.263433    0.843185
## std     16.382188      0.0      0.0  2.044410e-01    2.950492    1.556818
## min      4.605697    112.0      3.0  3.737686e-09    0.027819   -5.000000
## 25%     39.109745    112.0      3.0  8.092006e-02    0.688626    0.097830
## 50%     53.848823    112.0      3.0  1.561955e-01    1.167323    0.667966
## 75%     64.284463    112.0      3.0  2.838629e-01    2.383992    1.394105
## max     77.479132    112.0      3.0  1.000000e+00   15.000000    5.000000

Plots

Model parameters

Model predictions

Model 12: alpha_t, beta, pi_t

mod = 12
mod_info = print_model_info(mod)

## Model = model12
## Value function = rescorla_wagner_reinitV0
## Decision function = my_softmax_shrinking_press_bias
## Parameters = v0r, alpha, beta, pi_t

Free parameters: \(V0r\) = reinitialising \(V0\), \(\alpha_t\) = shrinking learning rate, \(\beta\) = inverse temperature, \(\pi_t\) = shrinking press bias

Parameter fits

model_folder, data_mod, data_mod_num = print_model_stats(mod)

##               nLL  Ntrials  Nparams
## count  170.000000    170.0    170.0
## mean    48.469944    112.0      4.0
## std     16.706184      0.0      0.0
## min      3.550610    112.0      4.0
## 25%     36.765907    112.0      4.0
## 50%     50.484815    112.0      4.0
## 75%     62.061840    112.0      4.0
## max     77.064211    112.0      4.0
##               v0r         alpha        beta        pi_t
## count  170.000000  1.700000e+02  170.000000  170.000000
## mean    -0.114130  1.145326e-01    3.334673    1.027638
## std      2.133012  1.404383e-01    3.832045    1.976753
## min     -5.000000  3.734932e-09    0.129691   -7.153210
## 25%     -1.135698  2.487038e-02    0.827480    0.182464
## 50%     -0.392928  6.628143e-02    1.693091    0.990258
## 75%      0.373953  1.509534e-01    4.570394    2.026127
## max      5.000000  1.000000e+00   15.000000    7.605972

Plots

Model parameters

Model predictions

Model comparison

Formulas

\[ BIC = 2 \cdot nLL + Nparams \cdot ln(Ntrials) \] \[ AIC = 2 \cdot nLL + 2 \cdot Nparams \] With: nLL = negative log likelihood
\[\\[.01in]\]

Results

	nLL	Ntrials	Nparams	BIC	AIC
model
0	58.107500	112	2	125.651998	120.215000
1	52.360935	112	3	118.877366	110.721870
2	50.262414	112	4	119.398823	108.524828
3	50.796260	112	4	120.466516	109.592520
4	52.822755	112	4	124.519506	113.645510
5	51.755054	112	3	117.665605	109.510108
6	46.913267	112	5	117.419029	103.826535
7	49.086785	112	5	121.766065	108.173571
8	47.798468	112	4	114.470932	103.596937
9	48.435748	112	4	115.745491	104.871496
10	49.505559	112	4	117.885113	107.011118
11	51.128443	112	3	116.412382	108.256885
12	48.469944	112	4	115.813884	104.939888
13	50.470229	112	4	119.814453	108.940458

Model comparison

Compare mod8 and mod9

##          nLL_mod8     v0_mod8    alpha_mod8   beta_mod8   pi_t_mod8
## count  170.000000  170.000000  1.700000e+02  170.000000  170.000000
## mean    47.798468   -0.116659  1.811305e-01    3.293070    1.062610
## std     16.920126    2.272556  2.089355e-01    3.500643    1.764987
## min      3.325319   -5.000000  1.264101e-16    0.122773   -6.415641
## 25%     37.156746   -1.234541  3.692270e-02    0.877272    0.192999
## 50%     50.614676   -0.338046  1.023679e-01    1.757615    0.948407
## 75%     61.386058    0.635814  2.453132e-01    4.711434    1.927004
## max     76.705921    5.000000  1.000000e+00   15.000000    6.668164

##          nLL_mod9     v0_mod9    alpha_mod9   beta_mod9   pi_t_mod9
## count  170.000000  170.000000  1.700000e+02  170.000000  170.000000
## mean    48.435748   -0.092623  1.139156e-01    3.445958    1.030806
## std     16.706529    2.170045  1.427369e-01    3.950705    2.004097
## min      3.563870   -5.000000  3.737686e-09    0.038199   -7.720410
## 25%     36.917690   -1.136027  2.639210e-02    0.818287    0.100299
## 50%     50.649393   -0.309904  6.618056e-02    1.717946    0.984759
## 75%     61.727837    0.472413  1.488237e-01    4.586511    1.948065
## max     77.041330    5.000000  1.000000e+00   15.000000    8.429271

Plots

Likelihoods

Parameters

Stats: t-tests \[V_0:\]

## mod8:  Ttest_1sampResult(statistic=-0.6693136632195669, pvalue=0.504208757753263)

## mod9:  Ttest_1sampResult(statistic=-0.5565136129805143, pvalue=0.5785959469927398)

Stats: paired t-tests

## nLL:
## means:
## mod8:  47.79846829164395
## mod9:  48.43574797474338
## normality asumption:
## mod8:  ShapiroResult(statistic=0.9671643376350403, pvalue=0.00047616203664802015)
## mod9:  ShapiroResult(statistic=0.965816855430603, pvalue=0.00034187137498520315)
## paired t-test:
## Ttest_relResult(statistic=-3.8856065639237953, pvalue=0.00014633609210649466)
## Wilcoxon:
## WilcoxonResult(statistic=4761.0, pvalue=9.61605197477806e-05)
## 
## 
## alpha:
## means:
## mod8:  0.18113047681349354
## mod9:  0.11391557796897692
## normality asumption:
## mod8:  ShapiroResult(statistic=0.766213595867157, pvalue=3.5342551368139352e-15)
## mod9:  ShapiroResult(statistic=0.6862607002258301, pvalue=1.4018962722958882e-17)
## paired t-test:
## Ttest_relResult(statistic=6.986333592510897, pvalue=6.205745788256391e-11)
## Wilcoxon:
## WilcoxonResult(statistic=1553.0, pvalue=6.018819096847661e-19)
## 
## 
## beta:
## means:
## mod8:  3.2930702178297024
## mod9:  3.4459583450832176
## normality asumption:
## mod8:  ShapiroResult(statistic=0.7934234142303467, pvalue=3.1690155463974176e-14)
## mod9:  ShapiroResult(statistic=0.7512106895446777, pvalue=1.1391245299393021e-15)
## paired t-test:
## Ttest_relResult(statistic=-0.8245677093453079, pvalue=0.4107793372206133)
## Wilcoxon:
## WilcoxonResult(statistic=6077.0, pvalue=0.06396807304546269)
## 
## 
## pi_t:
## means:
## mod8:  1.0626103441929362
## mod9:  1.0308060195981652
## normality asumption:
## mod8:  ShapiroResult(statistic=0.9398530125617981, pvalue=1.3971595080874977e-06)
## mod9:  ShapiroResult(statistic=0.9111093282699585, pvalue=1.2255516601555883e-08)
## paired t-test:
## Ttest_relResult(statistic=0.45381132580971545, pvalue=0.6505468573621205)
## Wilcoxon:
## WilcoxonResult(statistic=6366.0, pvalue=0.16069971758461887)
## 
## 
## v0:
## means:
## mod8:  -0.11665941953886272
## mod9:  -0.09262325863891928
## normality asumption:
## mod8:  ShapiroResult(statistic=0.9229835867881775, pvalue=7.624829123642485e-08)
## mod9:  ShapiroResult(statistic=0.919902503490448, pvalue=4.672326170407359e-08)
## paired t-test:
## Ttest_relResult(statistic=-0.2631533724384663, pvalue=0.7927529234474232)
## Wilcoxon:
## WilcoxonResult(statistic=6258.0, pvalue=0.14670780158044705)

Parameter recovery

Simulation
Parameter values for simulation sampled from fitted parameter values (numpy.random.normal function): \[ param \sim \mathcal{N}(mean, \, std) \]
Dataset was randomly chosen from the 170 datasets (random.suffle function).

Model 8

## Model = model8
## Value function = rescorla_wagner_shrinking_alpha
## Decision function = my_softmax_shrinking_press_bias
## Parameters = v0, alpha_t, beta, pi_t

N sim = 1000 Simulation parameter values:

##             v0   alpha_t      beta      pi_t
## mean -0.116659  0.181130  3.293070  1.062610
## std   2.272556  0.208935  3.500643  1.764987

Plots

Confusion matrix mod8

Conrrelations Model 8

Model 9

## Model = model9
## Value function = rescorla_wagner
## Decision function = my_softmax_shrinking_press_bias
## Parameters = v0, alpha, beta, pi_t

N sim = 1000 Simulation parameter values:

##             v0     alpha      beta      pi_t
## mean -0.092623  0.113916  3.445958  1.030806
## std   2.170045  0.142737  3.950705  2.004097

Plots

Confusion matrix Model 9

Conrrelations mod9